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Special Right Triangles Calculator.

Find missing sides of a 45-45-90 or 30-60-90 triangle.

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Results update as you type.

Result: leg: 5.00000000 · hypotenuse: 7.07106781

Result

leg: 5.00000000 · hypotenuse: 7.07106781

Use Cases

Geometry homework help

Quickly verify answers for problems involving special right triangles, ensuring correct side lengths and angle measures.

Example: Check that a 45-45-90 triangle with hypotenuse 10 has legs of 5√2.

Construction and design

Use special right triangle ratios to calculate diagonal lengths or dimensions in projects that involve 45° or 30°/60° angles.

Example: Find the height of a ramp that forms a 30-60-90 triangle with a 12-foot base.

Frequently Asked Questions

What are the special right triangles?
Special right triangles are 45-45-90 and 30-60-90 triangles. Their side lengths have fixed ratios, so knowing one side lets you find the others. For 45-45-90, the legs are equal and the hypotenuse is leg times √2. For 30-60-90, the sides are in ratio 1:√3:2.
How do I use the calculator?
Select the triangle type (45-45-90 or 30-60-90) and enter the length of any one side (leg or hypotenuse). The calculator will compute the missing side lengths and the angles (which are fixed for each type).
Can I enter the hypotenuse instead of a leg?
Yes. The calculator accepts any known side. For a 45-45-90 triangle, if you enter the hypotenuse, it divides by √2 to find the legs. For a 30-60-90 triangle, it uses the appropriate ratio to find the other sides.

Tips & Common Mistakes

Tips

  • Remember: in a 45-45-90 triangle, the legs are equal, and the hypotenuse is leg × √2.
  • For 30-60-90 triangles, the side opposite 30° is half the hypotenuse, and the side opposite 60° is √3 times the shorter leg.
  • Enter the known side exactly as given (e.g., 5 or 5√2) to get precise results.
  • Use the calculator to check your work, but also practice deriving the ratios manually.

Common Mistakes to Avoid

  • Confusing the ratios: 45-45-90 uses 1:1:√2, while 30-60-90 uses 1:√3:2.
  • Entering the hypotenuse as a leg, or vice versa, leading to incorrect calculations.
  • Forgetting that the angles are fixed: 45-45-90 has angles 45°, 45°, 90°; 30-60-90 has 30°, 60°, 90°.

Last updated: August 13, 2026